Soft core fluid in a quenched matrix of soft core particles: A mobile mixture in a model gel

We present a density-functional study of a binary phase-separating mixture of soft core particles immersed in a random matrix of quenched soft core particles of larger size. This is a model for a binary polymer mixture immersed in a cross-linked rigid polymer network. Using the replica “trick” for quenched-annealed mixtures we derive an explicit density functional theory that treats the quenched species on the level of its one-body density distribution. The relation to a set of effective external potentials acting on the annealed components is discussed. We relate matrix-induced condensation in bulk to the behavior of the mixture around a single large particle. The interfacial properties of the binary mixture at a surface of the quenched matrix display a rich interplay between capillary condensation inside the bulk matrix and wetting phenomena at the matrix surface.

Figure 1

The bulk phase diagram for a binary mixture of GCM particles with ${ϵ}_{12}∕{ϵ}_{11}=0.944$ and $R_{22}∕R_{11}=0.665$ (see also Ref. 21) in terms of the total density, ${\rho }^b$, and the concentration, $x$, of the smaller species 2. In (a) the paths $A_r$ and $B_r$ (arrows) are at constant total density in the reservoir. The paths $A_m$ and $B_m$ indicate the corresponding paths for the fluid in a matrix of density ${\rho }_0^b{R}_{11}^3={10}^{-3}$. (b) shows an expanded view of the phase diagram. The three additional pairs of binodal (solid line) and spinodal (dashed) lines show the loci in the reservoir phase diagram which correspond [via Eq. (27)] to the binodal/spinodal of the fluid adsorbed in the matrix, plotted for three different matrix densities. In (a) and (b) the solid line whose ends are denoted by filled circles is the thin-thick adsorbed film transition of the binary fluid adsorbed around a single big GCM particle with pair potential parameters ${ϵ}_{01}=k_{B}T$, ${ϵ}_{02}=0.8k_{B}T$, $R_{01}∕R_{11}=5.0$, and $R_{02}∕R_{11}=4.97$—see Refs. 26, 27. The two points $(◇)$ are where the spinodal for this single-particle thin-thick adsorbed transition meets the bulk fluid binodal.

Figure 2

Fluid densities, ${\rho }_n^b$, inside the matrix corresponding to a reservoir with fixed total density ${\rho }^b{R}_{11}^3=8.0$ and concentration $x=0.96$ plotted versus (uniform) matrix density ${\rho }_0^b$. For ${\rho }_0^b{R}_{11}^3>2.3\times {10}^{-3}$, condensation of the coexisting fluid phase occurs in the matrix. Note that ${\rho }_1^b$ is a nonmonotonic function of ${\rho }_0^b$, whereas ${\rho }_2^b$ and the total density ${\rho }^b$ are monotonically decreasing functions of ${\rho }_0^b$.

Figure 3

Density profiles of the binary GCM fluid at the planar interface between a matrix with density profile given by Eq. (36) and a reservoir of fluid (at $z>0$) with fixed total bulk density ${\rho }^b{R}_{11}^3=7.0$ and concentration $x=0.888$. The density profiles for species 1 are shown in (a) and those for species 2 are in (b). The density profiles are calculated for matrix bulk densities ${\rho }_0^b{R}_{11}^3={10}^{-5}$, ${10}^{-4}$, ${10}^{-3}$, ${10}^{-2}$, 0.05, and 0.1. Note that for intermediate densities the coexisting fluid phase is condensed in the matrix. At this reservoir state point, there is a wetting film extending from the interface at $z=0$ into the bulk reservoir fluid for the higher matrix densities.

Figure 4

Density profiles of the binary fluid at the planar interface between a matrix with density profile given by Eq. (36), with ${\rho }_0^b{R}_{11}^3={10}^{-3}$. The bulk fluid reservoir $(z\to \infty )$ has fixed total density ${\rho }^b{R}_{11}^3=7.0$. The profiles are calculated for concentrations $x=0.99$, 0.96, 0.94, 0.937, 0.934, 0.93, 0.91, 0.89, 0.886, and 0.885 45 (bulk coexistence is at $x_{coex}=0.88544$). These correspond to a series of points along path $A_r$ in Fig. 1a. Note that between $x=0.937$ and 0.934, condensation of the coexisting phase occurs in the matrix. Decreasing $x$ further, a wetting film develops on the surface of the matrix.

Figure 5

Density profiles of the binary fluid at the planar interface between a matrix with density profile given by Eq. (36), with ${\rho }_0^b{R}_{11}^3={10}^{-3}$. The bulk fluid reservoir $(z\to \infty )$ has total density ${\rho }^b{R}_{11}^3=6.0$. The profiles are calculated for concentrations $x=0.99$, 0.95, 0.92, 0.9, 0.89, 0.87, 0.84, 0.81, 0.8, 0.79, 0.788, 0.786, and 0.785 (bulk coexistence is at $x_{coex}=0.7848$). These correspond to a series of points along path $B_r$ in Fig. 1a. For decreasing $x$, a wetting film develops on the surface of the matrix. Note that, contrary to the case in Fig. 4, the density of the fluid adsorbed in the matrix changes continuously with $x$.